Glossary of real and complex analysis

This is a glossary of concepts and results in real analysis and complex analysis in mathematics.

See also: list of real analysis topics, list of complex analysis topics and glossary of functional analysis.

Abel
1.  Abel sum
2.  Abel integral
analytic continuation
An analytic continuation of a holomorphic function is a unique holomorphic extension of the function (on a connected open subset of  ).
argument principle
argument principle
Ascoli
Ascoli's theorem says that an equicontinous bounded sequence of functions on a compact subset of   has a convergent subsequence with respect to the sup norm.
Borel
1.  A Borel measure is a measure whose domain is the Borel σ-algebra.
2.  The Borel σ-algebra on a topological space is the smallest σ-algebra containing all open sets.
3.  Borel's lemma says that a given formal power series, there is a smooth function whose Taylor series coincides with the given series.
bump
A bump function is a nonzero compactly-supported smooth function, usually constructed using the exponential function.
Calderón
Calderón–Zygmund lemma
capacity
Capacity of a set is a notion in potential theory.
Carathéodory
Carathéodory's extension theorem
Cartan
Cartan's theorems A and B.
Cauchy
1.  The Cauchy–Riemann equations are a system of differential equations such that a function satisfying it (in the distribution sense) is a holomorphic function.
2.  Cauchy integral formula.
3.  Cauchy residue theorem.
4.  Cauchy's estimate.
5.  The Cauchy principal value is, when possible, a number assigned to a function when the function is not integrable.
6.  On a metric space, a sequence   is called a Cauchy sequence if  ; i.e., for each  , there is an   such that   for all  .
Cesàro
Cesàro summation is one way to compute a divergent series.
contour
The contour integral of a measurable function   over a piece-wise smooth curve   is  .
converge
1.  A sequence   on a topological space is said to converge to a point   if for each open neighborhood   of  , the set   is finite.
2.  A series   on a normed space (e.g.,  ) is said to converge if the sequence of the partial sums   converges.
convolution
The convolution   of two functions on a convex set is given by
 
provided the integration converges.
Cousin
Cousin problems.
cutoff
cutoff function.
Dedekind
A Dedekind cut is one way to construct real numbers.
derivative
Given a map   between normed spaces, the derivative of   at a point x is a (unique) linear map   such that  .
differentiable
A map between normed space is differentiable at a point x if the derivative at x exists.
differentiation
Lebesgue's differentiation theorem says:   for almost all x.
Dirac
The Dirac delta function   on   is a distribution (so not exactly a function) given as  
distribution
A distribution is a type of a generalized function; precisely, it is a continuous linear functional on the space of test functions.
divergent
A divergent series is a series whose partial sum does not converge. For example,   is divergent.
dominated
Lebesgue's dominated convergence theorem says   converges to   if   is a sequence of measurable functions such that   converges to   pointwise and   for some integrable function  .
edge
Edge-of-the-wedge theorem.
entire
An entire function is a holomorphic function whose domain is the entire complex plane.
equicontinuous
A set   of maps between fixed metric spaces is said to be equicontinuous if for each  , there exists a   such that   for all   with  . A map   is uniformly continuous if and only if   is equicontinuous.
Fatou
Fatou's lemma
Fourier
1.  The Fourier transform of a function   on   is: (provided it makes sense)
 
2.  The Fourier transform   of a distribution   is  . For example,   (Fourier's inversion formula).
Gauss
1.  The Gauss–Green formula
2.  Gaussian kernel
generalized
A generalized function is an element of some function space that contains the space of ordinary (e.g., locally integrable) functions. Examples are Schwartz's distributions and Sato's hyperfunctions.
Hardy
1.  The Hardy–Littlewood maximal inequality.
2.  Hardy space
Hartogs
1.  Hartogs extension theorem
2.  Hartogs's theorem on separate holomorphicity
harmonic
A function is harmonic if it satisfies the Laplace equation (in the distribution sense if the function is not twice differentiable).
Hausdorff
The Hausdorff–Young inequality says that the Fourier transformation   is a well-defined bounded operator when  .
Heaviside
The Heaviside function is the function H on   such that   and  .
Hilbert space
A Hilbert space is a real or complex inner product space that is a complete metric space with the metric induced by the inner product.
holomorphic function
A function defined on an open subset of   is holomorphic if it is complex differentiable. Equivalently, a function is holomorphic if it satisfies the Cauchy–Riemann equations (in the distribution sense if the function is not differentiable).
integrable
A measurable function   is said to be integrable if  .
integral
1.  The integral of the indicator function on a measurable set is the measure (volume) of the set.
2.  The integral of a measurable function is then defined by approximating the function by linear combinations of indicator functions.
Lebesgue integral
Lebesgue integral.
Lebesgue measure
Lebesgue measure.
Lelong
Lelong number.
Levi
Levi's problem asks to show a pseudoconvex set is a domain of holomorphy.
line integral
Line integral.
Liouville
Liouville's theorem says a bounded entire function is a constant function.
Lipschitz
1.  A map   between metric spaces is said to be Lipschitz continuous if  .
2.  A map is locally Lipschitz continuous if it is Lipschitz continuous on each compact subset.
maximum
The maximum principle says that a maximum value of a harmonic function in a connected open set is attained on the boundary.
measurable function
A measurable function is a structure-preserving function between measurable spaces in the sense that the preimage of any measurable set is measurable.
measurable set
A measurable set is an element of a σ-algebra.
measurable space
A measurable space consists of a set and a σ-algebra on that set which specifies what sets are measurable.
measure
A measure is a function on a measurable space that assigns to each measurable set a number representing its measure or size. Specifically, if X is a set and Σ is a σ-algebra on X, then a set-function μ from Σ to the extended real number line is called a measure if the following conditions hold:
  • Non-negativity: For all  
  •  
  • Countable additivity (or σ-additivity): For all countable collections   of pairwise disjoint sets in Σ,
 
measure space
A measure space consists of a measurable space and a measure on that measurable space.
meromorphic
A meromorphic function is an equivalence class of functions that are locally fractions of holomorphic functions.
method of stationary phase
The method of stationary phase.
microlocal
The notion microlocal refers to a consideration on the cotangent bundle to a space as opposed to that on the space itself. Explicitly, it amounts to considering functions on both points and momenta; not just functions on points.
Morera
Morera's theorem says a function is holomorphic if the integrations of it over arbitrary closed loops are zero.
Morse
Morse function.
Nevanlinna theory
Nevanlinna theory concerns meromorphic functions.
net
A net is a generalization of a sequence.
normed vector space
A normed vector space, also called a normed space, is a real or complex vector space V on which a norm is defined. A norm is a map   satisfying four axioms:
  1. Non-negativity: for every  , .
  2. Positive definiteness: for every  ,   if and only if   is the zero vector.
  3. Absolute homogeneity: for every scalar   and  , 
  4. Triangle inequality: for every   and  , 
Oka
Oka's coherence theorem says the sheaf   of holomorphic functions is coherent.
open
The open mapping theorem (complex analysis)
oscillatory integral
An oscillatory integral can give a sense to a formal integral expression like  
Paley
Paley–Wiener theorem
phase
The phase space to a configuration space   (in classical mechanics) is the cotangent bundle   to  .
plurisubharmonic
A function   on an open subset   is said to be plurisubharmonic if   is subharmonic for   in a neighborhood of zero in   and   points in  .
Poisson
Poisson kernel
power series
A power series is informally a polynomial of infinite degree; i.e.,  .
pseudoconex
A pseudoconvex set is a generalization of a convex set.
Radon measure
Let   be a locally compact Hausdorff space and let   be a positive linear functional on the space of continuous functions with compact support  . Positivity means that   if  . There exist Borel measures   on   such that   for all  . A Radon measure on   is a Borel measure that is finite on all compact sets, outer regular on all Borel sets, and inner regular on all open sets. These conditions guarantee that there exists a unique Radon measure   on   such that   for all  .
real-analytic
A real-analytic function is a function given by a convergent power series.
Riemann
1.  The Riemann integral of a function is either the upper Riemann sum or the lower Riemann sum when the two sums agree.
2.  The Riemann zeta function is a (unique) analytic continuation of the function   (it's more traditional to write   for  ).
3.  The Riemann hypothesis, still a conjecture, says each nontrivial zero of the Riemann zeta function has real part equal to  .
Runge
1.  Runge's approximation theorem.
2.  Runge domain.
Sato
Sato's hyperfunction, a type of a generalized function.
Schwarz
A Schwarz function is a function that is both smooth and rapid-decay.
semianalytic
The notion of semianalytic is an analog of semialgebraic.
sequence
A sequence on a set   is a map  .
series
A series is informally an infinite summation process  . Thus, mathematically, specifying a series is the same as specifying the sequence of the terms in the series. The difference is that, when considering a series, one is often interested in whether the sequence of partial sums   converges or not and if so, to what.
σ-algebra
A σ-algebra on a set is a nonempty collection of subsets closed under complements, countable unions, and countable intersections.
Stieltjes
Stieltjes–Vitali theorem
Stone–Weierstrass theorem
The Stone–Weierstrass theorem is any one of a number of related generalizations of the Weierstrass approximation theorem, which states that any continuous real-valued function defined on a closed interval can be uniformly approximated by polynomials. Let   be a compact Hausdorff space and let   have the uniform metric. One version of the Stone–Weierstrass theorem states that if   is a closed subalgebra of   that separates points and contains a nonzero constant function, then in fact  . If a subalgebra is not closed, taking the closure and applying the previous version of the Stone–Weierstrass theorem reveals a different version of the theorem: if   is a subalgebra of   that separates points and contains a nonzero constant function, then   is dense in  .
subanalytic
subanalytic.
subharmonic
A twice continuously differentiable function   is said to be subharmonic if   where   is the Laplacian. The subharmonicity for a more general function is defined by a limiting process.
subsequence
A subsequence of a sequence is another sequence contained in the sequence; more precisely, it is a composition   where   is a strictly increasing injection and   is the given sequence.
support
1.  The support of a function is the closure of the set of points where the function does not vanish.
2.  The support of a distribution is the support of it in the sense in sheaf theory.
Tauberian
Tauberian theory is a set of results (called tauberian theorems) concerning a divergent series; they are sort of converses to abelian theorems but with some additional conditions.
Taylor
Taylor expansion
tempered
A tempered distribution is a distribution that extends to a continuous linear functional on the space of Schwarz functions.
test
A test function is a compactly-supported smooth function.
uniform
1.  A sequence of maps   from a topological space to a normed space is said to converge uniformly to   if  .
2.  A map between metric spaces is said to be uniformly continuous if for each  , there exist a   such that   for all   with  .
Vitali
Vitali covering lemma.
Weierstrass
1.  Weierstrass preparation theorem.
2.  Weierstrass M-test.
Weyl
1.  Weyl calculus.
2.  Weyl quantization.
Whitney
1.  The Whitney extension theorem gives a necessary and sufficient condition for a function to be extended from a closed set to a smooth function on the ambient space.
2.  Whitney stratification

References

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  • Grauert, Hans; Remmert, Reinhold (1984). Coherent Analytic Sheaves. Springer.
  • Halmos, Paul R. (1974) [1950], Measure Theory, Graduate Texts in Mathematics, vol. 18, New York, Heidelberg, Berlin: Springer-Verlag, ISBN 978-0-387-90088-9, MR 0033869, Zbl 0283.28001
  • Hörmander, Lars (1983), The analysis of linear partial differential operators I, Grundl. Math. Wissenschaft., vol. 256, Springer, doi:10.1007/978-3-642-96750-4, ISBN 3-540-12104-8, MR 0717035.
  • Hörmander, Lars (1966). An Introduction to Complex Analysis in Several Variables. Van Nostrand.
  • Rudin, Walter (1976). Principles of Mathematical Analysis. Walter Rudin Student Series in Advanced Mathematics (3rd ed.). McGraw-Hill. ISBN 9780070542358.
  • Rudin, Walter (1986). Real and Complex Analysis (International Series in Pure and Applied Mathematics). McGraw-Hill. ISBN 978-0-07-054234-1.

Further reading

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